The long homology sequence for quasi-Banach spaces, with applications
Saying that \(0\to Y\to X\to Z\to 0\) is an exact sequence of quasi-Banach spaces means that \(X\) contains a subspace isomorphic to \(Y\) whose quotient is isomorphic to \(Z\). One also says that \(X\) is a twisted sum of \(Y\) and \(Z\), or an extension of \(Z\) by \(Y\). The class of all such extensions, modulo a natural equivalence relationship, is denoted \(\text{Ext}(Z,Y)\). It is well-known to algebraists that \(\text{Ext}(Z,Y)\) carries a vector space structure. The authors show that it also admits a complete semi-quasi-norm; in general, the resulting topology need not be Hausdorff. Their major result is that if \(E\) is another quasi-Banach space, then there is an exact sequence \(0\to L(Z,E)\to L(X,E)\to L(Y,E)\to\text{Ext}(Z,E)\to \text{Ext}(X,E)\to\text{Ext}(Y,E)\). The dual result for maps from (rather than into) \(E\) is also proved. The consequences of this are numerous, so we mention just a few which concern \(K\)-spaces, i.e., quasi-Banach spaces \(X\) for which \(\text{Ext}(X,E)\) is trivial whenever \(E\) is one-dimensional. One is a simple proof of the known result that quotients of Banach \(K\)-spaces are again \(K\)-spaces. Another is that the property \(\text{Ext}(\cdot,E)=0\) is a three-space property, and hence every twisted sum of an \({\mathcal L}_2\) space and an \({\mathcal L}_\infty\) space is a \(K\)-space.
scientific article; zbMATH DE number 2175434
scientific article